Mathematics • Year 10 • Unit 2 • Lesson 2
Expanding Brackets, Skill Drill
Build fluency with the distributive law a(b + c) = ab + ac. Multiply the outside through every inner term, keep the signs straight, and watch the negative multiplier. One worked example, one guided trace, eight independent problems.
1. I do, fully worked example
This one features a negative multiplier, the most common place to drop a sign. Read every step.
Problem. Expand −2(3y − 4).
Step 1, Identify the multiplier and inner terms.
Multiplier: −2. Inner terms: 3y and −4.
Reason: the multiplier is whatever sits outside the bracket, and it brings its sign with it.
Step 2, Multiply the multiplier by each inner term, signs and all.
(−2) × (3y) + (−2) × (−4)
Reason: distributive law, every inner term gets multiplied by the outside.
Step 3, Evaluate each product (mind the signs).
(−2)(3y) = −6y and (−2)(−4) = +8
Reason: negative × positive = negative; negative × negative = positive.
Step 4, Write the expanded form.
= −6y + 8
Reason: there are no like terms to combine, so this is the simplest form.
Answer: −2(3y − 4) = −6y + 8.
2. We do, fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. Expand and simplify 3(x + 2) − 2(x − 4).
Step 1, Expand the first bracket: 3(x + 2) = 3 × x + 3 × 2 = ____ + ____.
Step 2, Expand the second bracket. The multiplier is −2 (the minus sign comes WITH the 2):
−2(x − 4) = (−2) × x + (−2) × (−4) = ____ + ____
Step 3, Combine the two expansions:
3x + 6 + (____) + (____)
Step 4, Collect like terms (x-group, then constants):
x-group: 3x + ____ = ____ x
constants: 6 + ____ = ____
Step 5, Write the simplified expression:
Final answer = ____________________
3. You do, independent practice
Show your working. First four are foundation. Next two are standard (two brackets with simplification). Last two are extension.
Foundation, single bracket
3.1 Expand 4(x + 3). 1 mark
3.2 Expand 5(2x − 1). 1 mark
3.3 Expand −3(y + 4). 1 mark
3.4 Expand x(x + 5). 1 mark
Standard, expand AND simplify
3.5 Expand and simplify 5(2x − 3) + x. 2 marks
3.6 Expand and simplify 4(x + 1) − 3(x − 2). 2 marks
Extension, push your thinking
3.7 Expand and simplify 2x(3x − 4) + x(x + 5). 3 marks
3.8 A student writes 3(x + 4) = 3x + 4. In one sentence, say what the student forgot, then write the correct expansion. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (3(x + 2) − 2(x − 4))
Step 1: 3 × x + 3 × 2 = 3x + 6.
Step 2: −2(x − 4) = −2x + 8.
Step 3: 3x + 6 + (−2x) + (+8).
Step 4: x-group: 3x + (−2x) = 1x = x. Constants: 6 + 8 = 14.
Step 5: x + 14.
3.1-4(x + 3)
= 4(x) + 4(3) = 4x + 12.
3.2-5(2x − 1)
= 5(2x) + 5(−1) = 10x − 5.
3.3, −3(y + 4)
= (−3)(y) + (−3)(4) = −3y − 12.
Negative multiplier hits BOTH inner terms.
3.4, x(x + 5)
= x · x + x · 5 = x² + 5x.
x × x = x², not 2x.
3.5-5(2x − 3) + x
= 10x − 15 + x = 11x − 15.
3.6-4(x + 1) − 3(x − 2)
= 4x + 4 + (−3x + 6) = (4x − 3x) + (4 + 6) = x + 10.
The −3 multiplier turns −2 into +6, two negatives make a positive.
3.7-2x(3x − 4) + x(x + 5)
2x(3x − 4) = 6x² − 8x.
x(x + 5) = x² + 5x.
Combine: (6x² + x²) + (−8x + 5x) = 7x² − 3x.
3.8, The student forgot what?
The student only multiplied the 3 by the x and forgot to multiply it by the 4, the multiplier must hit every inner term.
Correct expansion: 3(x + 4) = 3x + 12.